feynman rules for reggeons - ICTP

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IC/67/70

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INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

FEYNMAN RULES FOR REGGEONS C. FRONSDAL

1967 PIAZZA OBERDAN

TRIESTE v-. "

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IC/67/7O

IHTERNATIONAL iTOMIC ENERGY AGMCY CEHTRE FOR THEORETICAL PHYSICS

RULES FOR REGGEONS * C. Pronsdal **

TRIESTE Septem"bsr 1967 *

To be submitted for publication. A preliminaxy report on this work was presented at the International Theoretical Physios Conference on Partioles and Fields, Rochester, 1967* Supported in part "by the MTational Science Foundation.

** On leave of absence from the University of California, Los .Angeles, USA.

ABSTRACT

Rules are formulated for the ©valuation of Feymnan diagrams in -which, the virtual lines represent infinite multiplets with discrete and/or

oontinuous spectra.

Reggeon Feynman diagrams.

In the simplest case this is a theory of

The fact that infinite multipletB can re-

present multi-particle states with oontinuous mass spectra is emphasized, and a special case of "Compton" scattering via two-particle intermediary states is studied in detail.

The kinematical structure of the

amplitudes is fixed by the vertices, and is physically reasonable in all channels, partioularly near u = 0 in unequal mass scattering.

This

allows a particularly convenient empirical representation of the dynamics. It should be emphasized that the mass spectrum of the infinite raultiplets are perfectly general, and that no higher symmetry is implied.

FEYNKAN" RULES FOR REGGEOUS I.

INTRODUCTION It is possible to conceive of physical states as members

of infinite multlplets. However, there is no reason to expect this to "be profitable if the states have to be mutilated in order to fit into a framework that is too narrow. Barly work on infinite multiplete idealized the Btatea to the point where they became an infinite set of one—partiole states with equal masses.

In thia case it ia possible to invoke invarianoe

under various non-compact groups to obtain predictions for oertain form factors*

' The convenient notation of group representations

can also be employed to construct explicit solutions of ourrent 2) 3) algebra ' and superconvergence relations. ' The next step is to consider one—particle states with unequal masses.

One prinoipal purpose of this report is to calculate

amplitudes related to the exohange of such objects.

' It is found

that both Regge pole contributions and Lorentz pole contributions appear.

The behaviour of an exchange amplitude is studied in detail

in Seo. IV.

It turns out that the vertex functions suggested by infinite

component local field theory (see below) gives a roaaonablc kineinatical structure', in particular, this is true near the infamous point u = 0. The physical states that occur in strong interaction scattering processes may be approximated by singlo-particle states but

poorly|

even if the masses are unequal*

—

Another main topic

of this paper is a consideration of multiplets that include multipartiole states. It turns out that the inclusion of multiparticle states does not create essential complications. Our approach is semi-empirical.

All our scattering amp-

litudes are related to simple Feynman diagrams, in which

the external

lines are one*"particle states, while each internal line represents an infinite multiplet. To define an amplitude it is necessary to speoify (i) the vertex functions and (ii) the propagators. 2) pletely empirioal theory

A com-

' would leave both arbitrary a priori,

and attempt the calculation of vertex functions and propagators from general physical requirements and additional assumptions like current algebra or superconvergence.

An example of the opposite extreme is

infinite-ooraponent local field theory, in which everything ie deduced -1-

from the Lagrangian or from a field equation. •^'

J

The advantage

of the empirical approaoh ia that a wide range of physical requirements can he introduced as input. Infinite-oomponent field theory, on the other hand, is a soluble model, and exact aoluloility is often very useful.

We shall therefore steer a middle course, which may "be

desoribed as searching for the physical world among the widest pos* sible olaas of soluble models.

Briefly, the idea is to leave pro-

pagators arbitrary, to be related directly to experiment, but to acoept the vertex functions suggested by infinite component field theory. Let the infinite multiplet be associated with an infinite set of fields, Y") , -i/r£z> if), -*• a r e known, the vertex function for the elementary interaction "between three states 11> , I 2 > , ! 3 > is given by

rc The simplest case is obtained by supposing that one of the fields is an ordinary scalar one-oomponent fields Then the vertex funotion reduces to

with t - ( 'b, - 'JJ ) . Suppose that the states \1> and \2y ar© identical except that "j). / -j> , then V1;?-i is the scalar form factor K ^ t ) of this state. Clearly XL (0) » 1 and ^ ( t ) ~> 0 as t -> -oO , because the overlap between the two states is perfect when -jx » j) 2 and vanishes in the limit of infinite momentum transfer. Hotice that the vertex funotions have been defined by the above for physical mass-shell states only. Off-mass-shell vertex funotions will be defined in terms of the Feynman amplitudes to whioh they contribute. The propagators are denoted L and are assumed to be invariant under Poinoare transformations. Acting on the fields V^in momentum space they are some complicated matrices!

However, these matrioes can be diagonalized. Let n be a set of Poincare-invariant quantum numbers. For example, ' if G « SO(3 n is just the spin i , defined invariantly by -3-

in terms of the Poincare generators. The physical states are eigenstates of these quantum numbers. We suppose that they form a complete set, so that the physical states may be uniquely labelled by -t>. and ft • Then L is diagonal in this basis,

The empirical input into our models is represented "by the arbitrary function L n (p)# The singularities of this function determine the mass speotrum of the physical states of the model, as well as the Kegge trajectories. If G o SO(3,1), then there can be only a discrete set of states, one for each value of the spin. But for larger groups the mass spectrum can have both a discrete and a continuous part. An example is discussed in Section II.3. It is known that the continuous part of the spectrum is capable of representing multipartiole states exaotly in the non-relativistic limit. After fixing the manifold of singularities of the propagator BO as to obtain the desired mass spectrum and Regge trajectories, we still have the residues of the singularities at our disposal. The residues form the absorbtive part of the propagator and are related to the completeness relation,

yr • Aks Z~' • V

=

1

We believe that assumptions of current algebra and superoonvergenoe can be formulated as conditions on the residue function, but this is not attempted in the present paper.

II.

INFIMITE KULTIPIETS AS INTERMEDIARY STATES

1,

"Cgmpton" scattering Kith single particle speotrum We shall oalculate the amplitude that corresponds to the Feynman

diagram of Fig. 1, in which the straight lines represent Reggeons, i.e. particles "belonging to an infinite rcultiplet, and the wavy lines are conventional scalar bosons, described lay a field A(x)»

In this

first example the infinite multiplet is of the simplest possible kindi

a single irreduoible representation of the homogeneous Lorentz

group„ Consider the irreducible representation I>(lO of SO(3,1) that is realized on the generalised tensor field

where the indices are four-^vector indices* unitary if (flf + l ) < 1 ;

This representation i s

for definiteness we shall take IT real, with

- 1< BT < 0

(II.2)

Then DfH") belongs to the supplementary series of unitary representations*

The vertioes are defined by the local invariant density

The amplitude is

A, - Z Here L

is the propagator for the virtual states, and the sura is

over a complete

' set of intermediary states*

choose definite states for the external lines;

It is necessary to this will be done

by replaoing , ), *••» we */(i>/)» For r>,'/* w1"^6 'fclie obvious analogue of the expansion (II.7)« Inserting these substitutions into (11,19) on© obtains (details are given in the appendix):

where

5.«QIGLfl)

(11.21)

and

The ooeffioient is the polynomial

b

p + - [(t-i)U (M+i)!lT ?

with

C4A)

(11.23)

and the oentre-of-mass "mean" velooity i s given lay

* f sThe coefficient bfl .simplifies in the equal mass casei lt r

0 , (11.25)

if t is even? in this oase (11.22) is a simple hypergeoiaetrie series.

' *

The sum over the intermediary states is evaluated just as before, and the final result is ta.J

(11.26)

3.

"Compton" scattering "with, multiparticle structure Both examples studied so far suffor from the defect that

the intermediary states are interpretable only as single-particle states, because every partial wave passes through a single intermediary state.

It is possible to enlarge the set of intermediary

states so that multiparticle intermediary states oan be taken into account exactly. -10-

In Subsection II. 1, let the group SO(3,1) "be replaced by SO(4,1), and consider the irreducible representation B(lT) of SO(4,1) that is realized on the generalized tensor field ^

« When X - "1» then Z > 1,

and the series does not converge unless L/>lc(e) decreases unreasonahly fast a s n - ^ o Q .

Hence (11.43) i s not applicable to the case of

spaoelike A . To olotain an expression for the amplitude that i s valid for values of P for which A i s spacelike, one may perform a Soinnierfeld-Watson transformation with respeot to the variable n. tlhen L^ is independent of n this can "be rigorously justified: The expansion (11.43) expresses a spherioal function for the -16-

representation D(W) of SO(4,1) in terms of the spherical funotions of the unitary representations D(n) of the subgroup G^, which is isomorphic to S0(4) when > is timelike. There exists

' another

expansion of the same function, in terms of the spherical funotions of the unitary representations of S0(3,l) —

or, more generally, the

subgroup G^ with spacelike A • The unitary representations of S0(3,l) that occur correspond to the ranges n - -1 + ij? , Q =» real, and -1 < n < 0

,

(II .49)

and the expansion can he written as an integral over these ranges of n, Furthermore, this integral converges for /\ => ± 1, and may he related to the sum (11.43) "by a Sommerfeld-Watson transformation when A «• +1. ' WG now postulate that the properties of L^ (s) are suoh as not to upset the possibility of performing the transformation, except that poles of L n (s) may l i e in the path of the deformation of the oontour, and have to be taken into account. The substitution of .the sum in (H.43)-by an integral over a hairpin oontour, and the subsequent deformation of the contour are completely straightforward. For dofiniteness let us suppose that iT (s) has the form

n - SQ , OH(e) is on tho line He c/(s) = - 1 . Thus oC(s) has a "branchpoint at the threshold a « s-. The physical sheet, cut from s « s n to +oO corresponds to He Of(s) > -lj the upper (lower) side of the cut corresponds to Im C>C(B) > 0 ( < 0). Write n - - 1 + xp for Im n > 0 and n = -1 - ip if Im n (_1 + 0((s) J maps the physical sheet of oomplex s on the -18-

oomplex plane cut from 0 to -00.

The bound state poles are con-

tained in the second term in (11.52), in the factor co3ec/T 0 We now turn to the evaluation of the amplitude A,, for

the Feynman diagram of Pig. 5. ^ e amplitudes A2,A , unlike A. ,A.» are not related by conventional a-u crossing* In the range (III.l) the calculation may "be carried out precisely as in Section II.2j one notices that V^ , defined by (11.24), varies from 0 to -1, so that the series (II.22) converges* The result is given "by (11*26), (11.27) after s is replaced "by u throughout. The "breakdown of conventional orossing symmetry oocurs in the same way as in the preceding disoussion of the annihilation amplitude A,.

3*

"C.onrpton" scattering .with u < 0 The expansions (11.13) and ( i l l . 5 ) are representations

of the amplitudes A^ and A, as sums

over the contributions of

irreducible representations of the l i t t l e group G^• the spin of the virtual s t a t e .

that i s , over

In both cases A was timelike, G\

was compact and the reduction (II.7) gave a discrete sum over spins. Turning now to the case of spacelike momentum transfer, we meet with the difficulty that (II.7) should be replaced by an integral over

Jl , since the range of / is now continuous.

It is doubtful

that the tensor method can be pushed as far as to give a useful formula of this type. tensor ^...^

An alternative is to begin by expanding tho

according to a discrete set of non-unitary represent-

ations of G^ , with n » IT, U-l, . * . .

However, since we already

have an expansion for the amplitude A^ for u > 0, the simplest -21-

procedure 18 to oontinue A^ analytically to negative u "by means of a Sommerfeld-Watson transformation. This i s the same procedure that was followed in Section I I . 3 . As in that case, we note that the transformation ia certainly possible, "barring complications due to unreasonable properties of the propagator. - 1 /(u)\ has the For definiteness, let ua suppose that L^ form

where ;gp(u) i s analytic in Re X > —•§• and sufficiently well behaved as X -> w», The physical mass spectrum is given "by the solution of KCmj2) - 9. ,

Hm 0 , 1 , 2 , . . ,

and Of(u) for negative u defines a Regge trajectory. troduce the abbreviation

Let us in

cm

-10)

Then the result of a completely straightforward Soomerfeld-Vatson transformation of (III.5) i s

A, -

(iii.il)

-22-

The third term is the contribution of the first of a string of fixed poles

' of the partial amplitude, at | =• II, 1T-1, . . . .

the approximation Lo (s) = L

In

(s) the first and the third term add

up to a four-dimensional spherical function and we recognize the Lorentz pole amplitude of Toller.

Indeed it is not surprising that

the amplitude fcr the exchange of an irreducible representation of the Lorentz group, in the equal mass case, should "be the same as a Lorentz pole contribution.

The second term in (III,11) is a dynamical

Regge pole contribution. Hotice that the Regge pole u, while the Lorentz pole is fixed.

moves with

The only interdependence of the

two is that their residues are equal in magnitude but of opposite sign in the case of coincidence, which occurs at the value of u for whioh D^(u) m IT. Thus for

0((u) near IT the amplitude is dominated

by a "Eegge dipole" with finite moment,

IV.

APPLICATIONS

1.

Kultipartiole states:

'

With the inclusion of multiparticle statsc in infinite niultiplets we hope that a principal obstacle to genuine physical applications has ~beQn removed.

It is known that an efficient treatment of the

non-relativistio Coulomb problem is feasible

'; probably the

relativistic pion-nuoleon problem should be attempted next. and KLEIMERT

3AROT

°' considered the I » -fr states, and found that a

group larger than S0(3,l) seems to be needed, because there are several isobars with the same spin and parity.

Here we have pointed

out that the representation of multiparticle states also requires large multiplets. Perhaps S0(4.l) or SU(2,2) is applicable to this problem -it is not hard to invent a propagator that corresponds to a single discrete state (the nucleon) and a oontinuum (nucleon + several pions) e

It seems natural to' expand the group in another

direction as well, to include isospin and strangeness and perhaps even SU(6).

This can be done by taking G =» SL(6,-C) or SU(6,6).

One of the objections against these groups has been the absence of experimental evidence for resonances with high isospin,'but if

-23-

continuous mass spectra are introduced then no resonances are needed. Kultipartiole states with arbitrarily high isospin certainly occur. 2.

Exchange amplitudest near u = 0 especially

Recently the behaviour of scattering amplitudes near u = 0 20) has "been studied by many authors . In particular, it has "been 21) pointed out that the customary analytic continuation from one channel to another can give an inconvenient representation of a scattering amplitude, due to the wrong or incomplete separation of the kinematioal factors.

Thus, in the u-channel, in which the

partial wave expansion is written down, kinematical

factors

appear which guarantee the correct threshold behaviour, and there is reason to hope that the partial wave amplitudes are smooth functions of u.

After analytic continuation to the s-channel, these

factors appear in the individual Regge pole contributions to the amplitude. Applications to experimental data have shown that, in this case, the rapid variation of the kinematical factors has to be cancelled by rapidly varying ~RQESQ residues.

The confusion

that has reigned recently; concerning the analytio structure of unequal mass scattering amplitudes near u = 0, is perhaps partly due to an inconvenient treatment of kinematical faotors.

The

amplitudes oaloulated in this paper may be regarded as models of Regge pole exchange.

As has been stressed repeatedly, the

"vertex funotions" must be regarded as "being essentially kinematioal. How we suggest that they are of a form that combines the correct threshold behaviour in one channel with the correot behaviour near u = 0 in the other. Consider the simplest case, the exchange of a minimal multiplet in "Compton" scattering.

The scattering amplitude corresponding

to the Peynman diagram of Pig. 3 was found to be given by the u-channel partial wave series

(«.)

-24-

(IV.1)

The series oonverges in both the relevant regions, u > (m- + m ) 2 and 0 < u < (m- - m 2 ) , but not uniformly near u a 0. The partial wave is

0?-17-1)1

In the u-ohannel, near the threshold u «• (m

2 + m ) ,

Thus (IV.2) contains the correct threshold factor; however, in other regions of the variable u, tho spherioal function behaves quite differently. u • 0.

Next, consider the positive neighbourhood of

Here, provided M" > - 1 ,

and

(iv.4)

-25-

In the equal mass oase, when iTa (u) is independent of *, the sum is equal to

L-1(u) (1 - cos6> f*ir\0)

u K (a+bsf

where a and "b are constants depending on the masses. of the faotor u~ in (IV.5).

(IV.5)

The singularity

in (IV.4) is exactly cancelled by the factor u

Thus it is seen that, in the case of trivial dynamics,

the kinematical structure does not give a singularity at u =* 0. fact we knew this already

In

since in the degenerate case the series

(iV.l) was summed exactly to (II.16). Finally, let us in'troduoe non-trivial dynamics.

If the function

L7 ( U ) does not have a singularity at u » 0, then the analytio struoture of the sum (IV.4) is given "by the "behaviour of the summand for large X . We may therefore approximate L | (u) by _ and A are linearly dependent, which i s guaranteed by momentum conservation if f «• 3$ f o r ^ / 3 i t i s a considerable loss of generality.

Then

S \K=

and thus .$

(A.12)

ThiB is to be raultipliod "by

21 (A.U) - * * /\

The syrametrization introduces a numerioal factor, equal to the probability that the 2IT-k A's in (A.12) hit tie This factor is ( . )

2F-T-2S

if t - r+2s and zero otherwise.

that remain are those with t - r+2a;

-30-

A'S

in (A.13),

The only torms

(A.14) where

/TV

(A.15) and

(A.16) In the case f - 3» which applies to Sec. II.2, we set

If ve use the conservation law 2P « p + p- and define s •* (p. + p^) , then

(A.18)

If instead 2P " p, - P2» "^en these formulae remain valid if s is replaced "by u « (p. - p-) . However, in this case (A.15) oonverges only if 0 < u < (m1 - m2)2 . Substituting (A.18) into (A.15)»\A.l6) veobtain (11.22), (11.23) of the main text. Finally, for the convenience of the reader, we write down the well known formula:

1

-j[ S-

-32-

t'